The Everton defender, 28, was given oxygen before being carried off following the challenge with Neil Taylor, who was sent off.
The incident happened midway through the second half of the important World Cup qualifier.
Republic manager Martin O'Neill said the defender suffered a "bad break".
"Seamus has gone to hospital, it's been confirmed by a doctor that he has broken his leg," added O'Neill.
"Obviously, it's a real blow to him. He's having the season of a lifetime at club level. He's a big player for us, a great captain and a great character.
"It's a big loss to Everton, a big loss to us. But he'll fight back I hope. It puts things in perspective."
Wales manager Chris Coleman said defender Taylor was "despondent" following the game.
"First and foremost, the most important thing is Seamus Coleman," he said. "We are told that it is not so good, which we are sorry for.
"Neil Taylor is not really that type of player, but it's a tough one for Seamus. Our thoughts are with him. I have not seen it again."
Everton return to Premier League action with the Merseyside derby against Liverpool at Anfield on Saturday, 1 April.
Everton midfielder James McCarthy was scheduled to start for the Republic, but was withdrawn from the team-sheet before kick-off due to a hamstring injury.
"He thought he was going to be OK with the couple of days (training) he had done," added O'Neill. "He was feeling it and I just didn't want to take any chances."
The draw in Dublin meant the Republic missed out on returning to the top of Group D, after Serbia beat Georgia earlier on Friday, with Wales four points behind in third.
Gareth Bale twice went close for Wales from long range, but the visitors had to withstand a spell of heavy pressure following Taylor's sending off.
Wales will also be without Real Madrid forward Bale when they visit Serbia on 11 June after he was booked for a foul on John O'Shea.
The Republic's next Group D qualifier is at home to Austria, also on 11 June.
